When Arcs Extend Uniquely: A Higher-Dimensional Generalization of Barlotti's Result
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912696029413376 |
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| author | Alderson, Tim L. |
| author_facet | Alderson, Tim L. |
| contents | In this short communication, we generalize a classical result of Barlotti concerning the unique extendability of arcs in the projective plane to higher-dimensional projective spaces. Specifically, we show that for integers \( k \ge 3 \), \( s \ge 0 \), and prime power \( q \), any \((n, k + s - 1)\)-arc in PG\((k - 1, q)\) of size \( n = (s+1)(q+1) + k - 3 \) admits a unique extension to a maximal arc, provided \( s + 2 \mid q \) and \( s < q - 2 \). This result extends the classical characterizations of maximal arcs in PG\((2,q)\) and connects naturally to the theory of A$^s$MDS codes. Our findings establish conditions under which linear codes of given dimension and Singleton defect can be uniquely extended to maximal-length projective codes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_06193 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | When Arcs Extend Uniquely: A Higher-Dimensional Generalization of Barlotti's Result Alderson, Tim L. Combinatorics Discrete Mathematics Primary: 94B65, Secondary: 94B25, 94B27 In this short communication, we generalize a classical result of Barlotti concerning the unique extendability of arcs in the projective plane to higher-dimensional projective spaces. Specifically, we show that for integers \( k \ge 3 \), \( s \ge 0 \), and prime power \( q \), any \((n, k + s - 1)\)-arc in PG\((k - 1, q)\) of size \( n = (s+1)(q+1) + k - 3 \) admits a unique extension to a maximal arc, provided \( s + 2 \mid q \) and \( s < q - 2 \). This result extends the classical characterizations of maximal arcs in PG\((2,q)\) and connects naturally to the theory of A$^s$MDS codes. Our findings establish conditions under which linear codes of given dimension and Singleton defect can be uniquely extended to maximal-length projective codes. |
| title | When Arcs Extend Uniquely: A Higher-Dimensional Generalization of Barlotti's Result |
| topic | Combinatorics Discrete Mathematics Primary: 94B65, Secondary: 94B25, 94B27 |
| url | https://arxiv.org/abs/2511.06193 |